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Approximate Inference using MCMC 9.520 Class 22 Ruslan Salakhutdinov BCS and CSAIL, MIT 1

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Page 1: Approximate Inference using MCMC - mit.edu9.520/spring10/Classes/class21_mcmc_2010.pdf · •In this class, we will concentrate on Markov Chain Monte Carlo (MCMC) methods for performing

Approximate Inferenceusing MCMC

9.520 Class 22

Ruslan SalakhutdinovBCS and CSAIL, MIT

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Page 2: Approximate Inference using MCMC - mit.edu9.520/spring10/Classes/class21_mcmc_2010.pdf · •In this class, we will concentrate on Markov Chain Monte Carlo (MCMC) methods for performing

Plan

1. Introduction/Notation.

2. Examples of successful Bayesian models.

3. Basic Sampling Algorithms.

4. Markov chains.

5. Markov chain Monte Carlo algorithms.

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Page 3: Approximate Inference using MCMC - mit.edu9.520/spring10/Classes/class21_mcmc_2010.pdf · •In this class, we will concentrate on Markov Chain Monte Carlo (MCMC) methods for performing

References/Acknowledgements

• Chris Bishop’s book: Pattern Recognition and Machine

Learning, chapter 11 (many figures are borrowed from this book).

• David MacKay’s book: Information Theory, Inference, and

Learning Algorithms, chapters 29-32.

• Radford Neals’s technical report on Probabilistic Inference Using

Markov Chain Monte Carlo Methods.

• Zoubin Ghahramani’s ICML tutorial on Bayesian Machine Learning:

http://www.gatsby.ucl.ac.uk/∼zoubin/ICML04-tutorial.html

• Ian Murray’s tutorial on Sampling Methods:

http://www.cs.toronto.edu/∼murray/teaching/

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Page 4: Approximate Inference using MCMC - mit.edu9.520/spring10/Classes/class21_mcmc_2010.pdf · •In this class, we will concentrate on Markov Chain Monte Carlo (MCMC) methods for performing

Basic Notation

P (x) probability of x

P (x|θ) conditional probability of x given θ

P (x, θ) joint probability of x and θ

Bayes Rule:

P (θ|x) =P (x|θ)P (θ)

P (x)

where

P (x) =

∫P (x, θ)dθ Marginalization

I will use probability distribution and probability density interchangeably. It should be obvious from the context.

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Page 5: Approximate Inference using MCMC - mit.edu9.520/spring10/Classes/class21_mcmc_2010.pdf · •In this class, we will concentrate on Markov Chain Monte Carlo (MCMC) methods for performing

Inference Problem

Given a dataset D = {x1, ..., xn}:

Bayes Rule:

P (θ|D) =P (D|θ)P (θ)

P (D)

P (D|θ) Likelihood function of θ

P (θ) Prior probability of θ

P (θ|D) Posterior distribution over θ

Computing posterior distribution is known as the inference problem.

But:

P (D) =

∫P (D, θ)dθ

This integral can be very high-dimensional and difficult to compute.

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Page 6: Approximate Inference using MCMC - mit.edu9.520/spring10/Classes/class21_mcmc_2010.pdf · •In this class, we will concentrate on Markov Chain Monte Carlo (MCMC) methods for performing

Prediction

P (θ|D) =P (D|θ)P (θ)

P (D)

P (D|θ) Likelihood function of θ

P (θ) Prior probability of θ

P (θ|D) Posterior distribution over θ

Prediction: Given D, computing conditional probability of x∗ requires

computing the following integral:

P (x∗|D) =

∫P (x∗|θ,D)P (θ|D)dθ

= EP (θ|D)[P (x∗|θ,D)]

which is sometimes called predictive distribution.

Computing predictive distribution requires posterior P (θ|D).

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Page 7: Approximate Inference using MCMC - mit.edu9.520/spring10/Classes/class21_mcmc_2010.pdf · •In this class, we will concentrate on Markov Chain Monte Carlo (MCMC) methods for performing

Model Selection

Compare model classes, e.g. M1 andM2. Need to compute posterior

probabilities given D:

P (M|D) =P (D|M)P (M)

P (D)

where

P (D|M) =

∫P (D|θ,M)P (θ,M)dθ

is known as the marginal likelihood or evidence.

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Page 8: Approximate Inference using MCMC - mit.edu9.520/spring10/Classes/class21_mcmc_2010.pdf · •In this class, we will concentrate on Markov Chain Monte Carlo (MCMC) methods for performing

Computational Challenges• Computing marginal likelihoods often requires computing very high-

dimensional integrals.

• Computing posterior distributions (and hence predictive

distributions) is often analytically intractable.

• In this class, we will concentrate on Markov Chain Monte Carlo

(MCMC) methods for performing approximate inference.

• First, let us look at some specific examples:

– Bayesian Probabilistic Matrix Factorization

– Bayesian Neural Networks

– Dirichlet Process Mixtures (last class)

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Page 9: Approximate Inference using MCMC - mit.edu9.520/spring10/Classes/class21_mcmc_2010.pdf · •In this class, we will concentrate on Markov Chain Monte Carlo (MCMC) methods for performing

Bayesian PMF

1234567...

1 2 3 4 5 6 7 ...

5 3 ? 1 ... 3 ? 4 ? 3 2 ...

~~R U

V

UserFeatures

FeaturesMovie

We have N users, M movies, and integer rating values from 1 to K.

Let rij be the rating of user i for movie j, and U ∈ RD×N , V ∈ RD×M

be latent user and movie feature matrices:

R ≈ U⊤V

Goal: Predict missing ratings.

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Page 10: Approximate Inference using MCMC - mit.edu9.520/spring10/Classes/class21_mcmc_2010.pdf · •In this class, we will concentrate on Markov Chain Monte Carlo (MCMC) methods for performing

Bayesian PMF

UVj i

Rij

j=1,...,Mi=1,...,N

σ

ΘV UΘ

ααV U Probabilistic linear model with Gaussianobservation noise. Likelihood:

p(rij|ui, vj, σ2) = N (rij|u

⊤i vj, σ

2)

Gaussian Priors over parameters:

p(U |µU ,ΛU) =N∏

i=1

N (ui|µu, Σu),

p(V |µV , ΛV ) =M∏

i=1

N (vi|µv, Σv).

Conjugate Gaussian-inverse-Wishart priors on the user and movie

hyperparameters ΘU = {µu,Σu} and ΘV = {µv,Σv}.

Hierarchical Prior.

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Page 11: Approximate Inference using MCMC - mit.edu9.520/spring10/Classes/class21_mcmc_2010.pdf · •In this class, we will concentrate on Markov Chain Monte Carlo (MCMC) methods for performing

Bayesian PMF

Predictive distribution: Consider predicting a rating r∗ij for user i

and query movie j:

p(r∗ij|R) =

∫∫p(r∗ij|ui, vj)p(U, V,ΘU ,ΘV |R)︸ ︷︷ ︸

Posterior over parameters and hyperparameters

d{U, V }d{ΘU ,ΘV }

Exact evaluation of this predictive distribution is analytically

intractable.

Posterior distribution p(U, V,ΘU ,ΘV |R) is complicated and does not

have a closed form expression.

Need to approximate.

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Page 12: Approximate Inference using MCMC - mit.edu9.520/spring10/Classes/class21_mcmc_2010.pdf · •In this class, we will concentrate on Markov Chain Monte Carlo (MCMC) methods for performing

Bayesian Neural NetsRegression problem: Given a set of i.i.d observations X = {xn}Nn=1

with corresponding targets D = {tn}Nn=1.

x0

x1

xD

z0

z1

zM

y1

yK

w(1)MD w

(2)KM

w(2)10

hidden units

inputs outputs

Likelihood:

p(D|X,w) =N∏

n=1

N (tn|y(xn,w), β2)

The mean is given by the outputof the neural network:

yk(x,w) =M∑

j=0

w2kjσ

( D∑

i=0

w1jixi

)

where σ(x) is the sigmoid function.

Gaussian prior over the network parameters: p(w) = N (0, α2I).

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Page 13: Approximate Inference using MCMC - mit.edu9.520/spring10/Classes/class21_mcmc_2010.pdf · •In this class, we will concentrate on Markov Chain Monte Carlo (MCMC) methods for performing

Bayesian Neural Nets

x0

x1

xD

z0

z1

zM

y1

yK

w(1)MD w

(2)KM

w(2)10

hidden units

inputs outputs

Likelihood:

p(D|X,w) =N∏

n=1

N (tn|y(xn,w), β2)

Gaussian prior over parameters:

p(w) = N (0, α2I)

Posterior is analytically intractable:

p(w|D,X) =p(D|w,X)p(w)∫p(D|w,X)p(w)dw

Remark: Under certain conditions, Radford Neal (1994) showed, as the number of

hidden units go to infinity, a Gaussian prior over parameters results in a Gaussian

process prior for functions.

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Page 14: Approximate Inference using MCMC - mit.edu9.520/spring10/Classes/class21_mcmc_2010.pdf · •In this class, we will concentrate on Markov Chain Monte Carlo (MCMC) methods for performing

Undirected Modelsx is a binary random vector with xi ∈ {+1,−1}:

p(x) =1

Zexp

( ∑

(i,j)∈E

θijxixj +∑

i∈V

θixi

).

where Z is known as partition function:

Z =∑

x

exp( ∑

(i,j)∈E

θijxixj +∑

i∈V

θixi

).

If x is 100-dimensional, need to sum over 2100 terms.

The sum might decompose (e.g. junction tree). Otherwise we need

to approximate.

Remark: Compare to marginal likelihood.

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Page 15: Approximate Inference using MCMC - mit.edu9.520/spring10/Classes/class21_mcmc_2010.pdf · •In this class, we will concentrate on Markov Chain Monte Carlo (MCMC) methods for performing

Monte Carlo

p(z) f(z)

z

For most situations we will be interestedin evaluating the expectation:

E[f ] =

∫f(z)p(z)dz

We will use the following notation: p(z) = p(z)Z .

We can evaluate p(z) pointwise, but cannot evaluate Z.

• Posterior distribution: P (θ|D) = 1P (D)P (D|θ)P (θ)

• Markov random fields: P (z) = 1Z exp(−E(z))

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Page 16: Approximate Inference using MCMC - mit.edu9.520/spring10/Classes/class21_mcmc_2010.pdf · •In this class, we will concentrate on Markov Chain Monte Carlo (MCMC) methods for performing

Simple Monte Carlo

General Idea: Draw independent samples {z1, ..., zn} from

distribution p(z) to approximate expectation:

E[f ] =

∫f(z)p(z)dz ≈

1

N

N∑

n=1

f(zn) = f

Note that E[f ] = E[f ], so the estimator f has correct mean (unbiased).

The variance:

var[f ] =1

NE

[(f − E[f ])2

]

Remark: The accuracy of the estimator does not depend on

dimensionality of z.

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Page 17: Approximate Inference using MCMC - mit.edu9.520/spring10/Classes/class21_mcmc_2010.pdf · •In this class, we will concentrate on Markov Chain Monte Carlo (MCMC) methods for performing

Simple Monte Carlo

In general:∫

f(z)p(z)dz ≈1

N

N∑

n=1

f(zn), zn ∼ p(z)

Predictive distribution:

P (x∗|D) =

∫P (x∗|θ,D)P (θ|D)dθ

≈1

N

N∑

n=1

P (x∗|θn,D), θn ∼ p(θ|D)

Problem: It is hard to draw exact samples from p(z).

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Page 18: Approximate Inference using MCMC - mit.edu9.520/spring10/Classes/class21_mcmc_2010.pdf · •In this class, we will concentrate on Markov Chain Monte Carlo (MCMC) methods for performing

Basic Sampling Algorithm

How to generate samples from simple non-uniform distributions

assuming we can generate samples from uniform distribution.

p(y)

h(y)

y0

1

Define: h(y) =∫ y

−∞p(y)dy

Sample: z ∼ U [0, 1].

Then: y = h−1(z) is a sample from p(y).

Problem: Computing cumulative h(y) is just as hard!

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Page 19: Approximate Inference using MCMC - mit.edu9.520/spring10/Classes/class21_mcmc_2010.pdf · •In this class, we will concentrate on Markov Chain Monte Carlo (MCMC) methods for performing

Rejection Sampling

Sampling from target distribution p(z) = p(z)/Zp is difficult.

Suppose we have an easy-to-sample proposal distribution q(z), such

that kq(z) ≥ p(z), ∀z.

z0 z

u0

kq(z0)kq(z)

p(z)

Sample z0 from q(z).

Sample u0 from Uniform[0, kq(z0)]

The pair (z0, u0) has uniform distribution

under the curve of kq(z).

If u0 > p(z0), the sample is rejected.

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Page 20: Approximate Inference using MCMC - mit.edu9.520/spring10/Classes/class21_mcmc_2010.pdf · •In this class, we will concentrate on Markov Chain Monte Carlo (MCMC) methods for performing

Rejection Sampling

z0 z

u0

kq(z0)kq(z)

p(z)

Probability that a sample is accepted is:

p(accept) =

∫p(z)

kq(z)q(z)dz

=1

k

∫p(z)dz

The fraction of accepted samples depends on the ratio of the area

under p(z) and kq(z).

Hard to find appropriate q(z) with optimal k.

Useful technique in one or two dimensions. Typically applied as a

subroutine in more advanced algorithms.

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Page 21: Approximate Inference using MCMC - mit.edu9.520/spring10/Classes/class21_mcmc_2010.pdf · •In this class, we will concentrate on Markov Chain Monte Carlo (MCMC) methods for performing

Importance Sampling

Suppose we have an easy-to-sample proposal distribution q(z), such

that q(z) > 0 if p(z) > 0.

p(z) f(z)

z

q(z) E[f ] =

∫f(z)p(z)dz

=

∫f(z)

p(z)

q(z)q(z)dz

≈1

N

n

p(zn)

q(zn)f(zn), zn ∼ q(z)

The quantities wn = p(zn)/q(zn) are known as importance weights.

Unlike rejection sampling, all samples are retained.

But wait: we cannot compute p(z), only p(z).

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Page 22: Approximate Inference using MCMC - mit.edu9.520/spring10/Classes/class21_mcmc_2010.pdf · •In this class, we will concentrate on Markov Chain Monte Carlo (MCMC) methods for performing

Importance SamplingLet our proposal be of the form q(z) = q(z)/Zq:

E[f ] =

∫f(z)p(z)dz =

∫f(z)

p(z)

q(z)q(z)dz =

Zq

Zp

∫f(z)

p(z)

q(z)q(z)dz

≈Zq

Zp

1

N

n

p(zn)

q(zn)f(zn) =

Zq

Zp

1

N

n

wnf(zn), zn ∼ q(z)

But we can use the same importance weights to approximateZp

Zq:

Zp

Zq

=1

Zq

∫p(z)dz =

∫p(z)

q(z)q(z)dz ≈

1

N

n

p(zn)

q(zn)=

1

N

n

wn

Hence:

E[f ] ≈1

N

n

wn

∑n wn

f(zn) Consistent but biased.

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Page 23: Approximate Inference using MCMC - mit.edu9.520/spring10/Classes/class21_mcmc_2010.pdf · •In this class, we will concentrate on Markov Chain Monte Carlo (MCMC) methods for performing

ProblemsIf our proposal distribution q(z) poorly matches our target distribution

p(z) then:

• Rejection Sampling: almost always rejects

• Importance Sampling: has large, possibly infinite, variance

(unreliable estimator).

For high-dimensional problems, finding good proposal distributions is

very hard. What can we do?

Markov Chain Monte Carlo.

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Page 24: Approximate Inference using MCMC - mit.edu9.520/spring10/Classes/class21_mcmc_2010.pdf · •In this class, we will concentrate on Markov Chain Monte Carlo (MCMC) methods for performing

Markov ChainsA first-order Markov chain: a series of random variables {z1, ..., zN}

such that the following conditional independence property holds for

n ∈ {z1, ..., zN−1}:

p(zn+1|z1, ..., zn) = p(zn+1|zn)

We can specify Markov chain:

• probability distribution for initial state p(z1).

• conditional probability for subsequent states in the form of transition

probabilities T (zn+1←zn) ≡ p(zn+1|zn).

Remark: T (zn+1←zn) is sometimes called a transition kernel.

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Page 25: Approximate Inference using MCMC - mit.edu9.520/spring10/Classes/class21_mcmc_2010.pdf · •In this class, we will concentrate on Markov Chain Monte Carlo (MCMC) methods for performing

Markov ChainsA marginal probability of a particular state can be computed as:

p(zn+1) =∑

zn

T (zn+1←zn)p(zn)

A distribution π(z) is said to be invariant or stationary with respect

to a Markov chain if each step in the chain leaves π(z) invariant:

π(z) =∑

z′

T (z←z′)π(z′)

A given Markov chain may have many stationary distributions. For

example: T (z←z′) = I{z = z′} is the identity transformation. Then

any distribution is invariant.

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Page 26: Approximate Inference using MCMC - mit.edu9.520/spring10/Classes/class21_mcmc_2010.pdf · •In this class, we will concentrate on Markov Chain Monte Carlo (MCMC) methods for performing

Detailed BalanceA sufficient (but not necessary) condition for ensuring that π(z) is

invariant is to choose a transition kernel that satisfies a detailed

balance property:

π(z′)T (z←z′) = π(z)T (z′←z)

A transition kernel that satisfies detailed balance will leave that

distribution invariant:

z′

π(z′)T (z←z′) =∑

z′

π(z)T (z′←z)

= π(z)∑

z′

T (z′←z) = π(z)

A Markov chain that satisfies detailed balance is said to be reversible.

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Page 27: Approximate Inference using MCMC - mit.edu9.520/spring10/Classes/class21_mcmc_2010.pdf · •In this class, we will concentrate on Markov Chain Monte Carlo (MCMC) methods for performing

RecapWe want to sample from target distribution π(z) = π(z)/Z

(e.g. posterior distribution).

Obtaining independent samples is difficult.

• Set up a Markov chain with transition kernel T (z′←z) that leaves

our target distribution π(z) invariant.

• If the chain is ergodic, i.e. it is possible to go from every state to

any other state (not necessarily in one move), then the chain will

converge to this unique invariant distribution π(z).

• We obtain dependent samples drawn approximately from π(z) by

simulating a Markov chain for some time.

Ergodicity: There exists K, for any starting z, T K(z′←z) > 0 for all π(z′) > 0.

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Page 28: Approximate Inference using MCMC - mit.edu9.520/spring10/Classes/class21_mcmc_2010.pdf · •In this class, we will concentrate on Markov Chain Monte Carlo (MCMC) methods for performing

Metropolis-Hasting AlgorithmA Markov chain transition operator from current state z to a new

state z′ is defined as follows:

• A new ’candidate’ state z∗ is proposed according to some proposal

distribution q(z∗|z), e.g. N (z, σ2).

• A candidate state x∗ is accepted with probability:

min

(1,

π(z∗)

π(z)

q(z|z∗)

q(z∗|z)

)

• If accepted, set z′ = z∗. Otherwise z′ = z, or the next state is the

copy of the current state.

Note: no need to know normalizing constant Z.

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Page 29: Approximate Inference using MCMC - mit.edu9.520/spring10/Classes/class21_mcmc_2010.pdf · •In this class, we will concentrate on Markov Chain Monte Carlo (MCMC) methods for performing

Metropolis-Hasting AlgorithmWe can show that M-H transition kernel leaves π(z) invariant byshowing that it satisfies detailed balance:

π(z)T (z′←z) = π(z)q(z′|z)min

(1,

π(z′)

π(z)

q(z|z′)

q(z′|z)

)

= min (π(z)q(z′|z), π(z′)q(z|z′))

= π(z′) min

(π(z)

π(z′)

)q(z′|z)

q(z|z′), 1

)

= π(z′)T (z←z′)

Note that whether the chain is ergodic will depend on the particulars

of π and proposal distribution q.

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Page 30: Approximate Inference using MCMC - mit.edu9.520/spring10/Classes/class21_mcmc_2010.pdf · •In this class, we will concentrate on Markov Chain Monte Carlo (MCMC) methods for performing

Metropolis-Hasting Algorithm

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

Using Metropolis algorithm to sample

from Gaussian distribution with

proposal q(z′|z) = N (z, 0.04).

accepted (green), rejected (red).

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Page 31: Approximate Inference using MCMC - mit.edu9.520/spring10/Classes/class21_mcmc_2010.pdf · •In this class, we will concentrate on Markov Chain Monte Carlo (MCMC) methods for performing

Choice of Proposal

σmax

σmin

ρ

Proposal distribution:

q(z′|z) = N (z, ρ2).

ρ large - many rejections

ρ small - chain moves too slowly

The specific choice of proposal can greatly affect the performance of

the algorithm.

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Gibbs SamplerConsider sampling from p(z1, ..., zN).

z1

z2

L

l

Initialize zi, i = 1, ..., N

For t=1,...,T

Sample zt+11 ∼ p(z1|z

t2, ..., z

tN)

Sample zt+12 ∼ p(z2|z

t+11 , xt

3, ..., ztN)

· · ·

Sample zt+1N ∼ p(zN |z

t+11 , ..., zt+1

N−1)

Gibbs sampler is a particular instance of M-H algorithm with proposals

p(zn|zi6=n) → accept with probability 1. Apply a series (component-

wise) of these operators.

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Gibbs SamplerApplicability of the Gibbs sampler depends on how easy it is to sample

from conditional probabilities p(zn|zi6=n).

• For discrete random variables with a few discrete settings:

p(zn|zi6=n) =p(zn, zi6=n)∑zn

p(zn, zi6=n)

The sum can be computed analytically.

• For continuous random variables:

p(zn|zi6=n) =p(zn, zi6=n)∫

p(zn, zi6=n)dzn

The integral is univariate and is often analytically tractable or

amenable to standard sampling methods.

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Bayesian PMFRemember predictive distribution?: Consider predicting a rating

r∗ij for user i and query movie j:

p(r∗ij|R) =

∫∫p(r∗ij|ui, vj)p(U, V,ΘU ,ΘV |R)︸ ︷︷ ︸

Posterior over parameters and hyperparameters

d{U, V }d{ΘU ,ΘV }

Use Monte Carlo approximation:

p(r∗ij|R) ≈1

N

N∑

n=1

p(r∗ij|u(n)i , v

(n)j ).

The samples (uni , vn

j ) are generated by running a Gibbs sampler, whose

stationary distribution is the posterior distribution of interest.

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Page 35: Approximate Inference using MCMC - mit.edu9.520/spring10/Classes/class21_mcmc_2010.pdf · •In this class, we will concentrate on Markov Chain Monte Carlo (MCMC) methods for performing

Bayesian PMFMonte Carlo approximation:

p(r∗ij|R) ≈1

N

N∑

n=1

p(r∗ij|u(n)i , v

(n)j ).

The conditional distributions over the user and movie feature vectors

are Gaussians → easy to sample from:

p(ui|R,V,ΘU , α) = N(ui|µ

∗i ,Σ

∗i

)

p(vj|R,U,ΘU , α) = N(vj|µ

∗j ,Σ

∗j

)

The conditional distributions over hyperparameters also have closed

form distributions → easy to sample from.

Netflix dataset – Bayesian PMF can handle over 100 million ratings.

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Page 36: Approximate Inference using MCMC - mit.edu9.520/spring10/Classes/class21_mcmc_2010.pdf · •In this class, we will concentrate on Markov Chain Monte Carlo (MCMC) methods for performing

MCMC: Main ProblemsMain problems of MCMC:

• Hard to diagnose convergence (burning in).

• Sampling from isolated modes.

More advanced MCMC methods for sampling in distributions with

isolated modes:

• Parallel tempering

• Simulated tempering

• Tempered transitions

Hamiltonian Monte Carlo methods (make use of gradient information).

Nested Sampling, Coupling from the Past, many others.

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Page 37: Approximate Inference using MCMC - mit.edu9.520/spring10/Classes/class21_mcmc_2010.pdf · •In this class, we will concentrate on Markov Chain Monte Carlo (MCMC) methods for performing

Deterministic Methods

• Laplace Approximation

• Bayesian Information Criterion (BIC)

• Variational Methods: Mean-Field, Loopy Belief Propagation along

with various adaptations.

• Expectation Propagation.

• · · ·

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